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                                  5                           
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                             Z
                                by DrMAD

                                     -  ப...  ப...
                                        ப...

                                       ()  

    
    1.  ⮤
    2. ᪨ ⮤
    2.1. ⪠  - ᯮ樠쭮 ࠧ
    2.2. ᪠    
    2.3.    " 宫"
    2.4.   ⨥ ⨢
    2.5.   
    3. 樮 ⮤
    3.1. 饥 ᠭ 饣 
    3.2.  १
    3.3. 楯 : 訫  
    

                             

    ᫨     ⠥,    ᮢ६  ⥢  ,  
୥ ⪮ ମ,   ⮢ 騪 뢭  ⪨ 
⭨   㬮,  -  "-- ਪ쭮,  ",  ,
, ⭮,    ।⠢  ⮬,   
࠭ ᮢ  类  .   㦭  ⮣, ⮡
ࠢ쭮 ࣠ ⨢   ᥬ୮  ஢;  ⮡
⠢ ᢮  - ࠢ쭮   ࣠;   ,  ,
⮡  ࠧ 㬠, ⮨  ᪠ ᢮ 猪   In
The Wild.
    ࠤ樮,  ᫥    ᫮  ꥪ⮢    
ᯮ  㯯 ⮤:
    - ;
    - ᪨;
    - 樮.

                          1.  ⮤

    ᪨,  ७    ࠪ⨪    ॠ쭮
饬   㭪樮饬 ꥪ.  ᪮ "ꥪ⮬"  
砥   ୥,   ᤥ  ࠩ ᫮    
,   室 孨᪨  ࣠樮 .
     ⠪  ७  -⠪  .    ࢥ  㭨
ફ    ६      ஦
ᠬ 䨪.     㡫    㦠    ⢥
ࠢ 窨  ᠬ ࠧ 㬮祭.
       ,   ⫨    䨪    筮
  ⮫쪮    ⮬ 砥,  ᫨  TCP-  -
⪠,   ⫨   -.  直, ᯮ騥
-    _⥢_  ,  ⠪  .   shell-,
    த      .  ⮢  直,
ᯮ騥  ⮤ 樠쭮 ਨ  ન  _⮢_ ,
⠪ ⪨ ( ࠩ ,  㦨)  ,  ⮬
    ࠭ 묨 ⮤ ࠩ 㤭,   
   .
     ।,  ࠪ᪨  ᬮ७  ⥬᪨
 ஢७   ⭮  ⥬  ࠢ    १⮢  
१⠬    ᫥.         
㤮⢮⥫.

                        2. ᪨ ⮤

     ⮤,   ᭮      -   ⥮᪮  ,
।⠢      ࠢ,  ஥  뢠      ᮡ
    ࠪ⨪ ꥪ  .  宩
  ᪨    ࠭  ᮢ  
    /2/,     ⮩  ਢ ⮫쪮 ⮢ १.
ࠧ  ᠬ⥫쭮 ⠢ ࠢ   .
    ⠪,  ᫥騥 祭.
     N - 饥 ⢮ 設,  V - ⢮ ஢, W
- ⢮   ஢. ⢥, N = V+W. ⮡  
 N,   V   W      ᬠਢ    ࠦ  
஢ 設  饣 ⢠:  v=V/N  w=W/N,  ࠧ㬥 v+w=1.
஬ ⮣,      ,        ⠭,    㭪樨,
騥  ६: v= v(t)  w=w(t).

               2.1. ⪠  - ᯮ樠쭮 ࠧ

    ।,    ᢮   砩,  ࠧ  100
%- 䥪⨢ 㦨   ஢ 設    ࠦ
.       t=0 ࠦ ᥣ  設:  V(0)=1.   
। ᪮ ࠧ = B.
    ⢮ ࠦ   設   ﭭ   ⠥      
⢮ ࠭ ࠦ ᮡ쥢  אַ ய樮쭮  B. 
⥩襥 ७樠쭮 ࠢ:

                           dV(t)/dt = B*V(t)

     襭  V(0)=1 :

                            V(t)=exp(B*t).

       १      㣨 ᯮᮡ.   
ࠦ     ६  ࠦ  A  ᢮  ᮡ쥢.
 V(1)=A+1;  V(2)=A+1+(A+1)*A=(A+1)^2;   ⠪ :  V(t)=(A+1)^t.
஢ ਢ ८ࠧ, 稬:

    (A+1)^t = exp(ln(A+1)*t) =  exp(B*t),

     B=ln(A+1)   -   ⠪   䨧᪨      ।   ᪮
ࠧ.
      ( 䨧  뢠 "⪮ "  /1/) 뢠
 ᯮ樠쭮 ࠧ .

                              v ^     
                                |     
                                |    
                                |    
                                |   
                                | 
                                +---------------->  t

    ⢥ ⮨⢮ ⮩  -  .    ॠ쭮
  뢠...    ,  ⮡  ᮢᥬ  ࠢ쭮...    
ᥣ.

            2.2. "᪠"    

     ॠ쭮  ⢠  롨ࠥ  砩,  ⮬  ࠦ
設 ""  ⮫쪮  ஢,     㣠.  祬 
㤥 ࠦ 設,  ⥬    䥪⨢  ⠭  砩
"堭".
      ࠦ 設 -  ᪮   v(t), ..
ࢠ  ந   ६.  ᫮ ᮮࠧ,      ⮬
砥    ⮫쪮  אַ  ய樮쭠    㢥稢饩   
ࠦ,     㬥蠥   ਡ  ᢮ ।쭮
祭.
    ⠪,  ७樠쭮 ࠢ 1- 浪:

      dv(t)/dt = B*v(t)*(1-v(t)) = B*v(t)-B*v(t)^2.

         (ࠢ    )    뢠   
 ,  㦥  150 (  ६   
஢ ࠧ      ਯ  ).
     襭 - "᪠" ਢ :

                        v(t) = 1/(1+exp(-Bt)).

                              v ^      
                                |     
                                |    
                                |    
                                |   
                                | 
                                +---------------->  t

    ᬮਬ । 砨:
    1) v(t)      0,  w(t)    1,  ⮣   
  ""  ࠦ    設,          ᥡ   
ᮮ⢥⢨  "⪮ ";
    2) v(t)   1,  w(t)    0,        ""
    ࠭  㦥  ࠦ 設,     ࠦ
ࠩ ;
    3) v(t)    w(t)      0.5,      ࠦ 設
ᨬ, ᪮  筨 ࠧ ,   ⥭樠 
 筮.
     뢮:  ⥣  砩  롮  楫    ࠦ
(IP-)   -        ᠬ  ⨬쭠.      ,  
ᯮ짮  ࠪ᪨    ⥢  ,  맢訥  
   ᫥  5  ,      ⮬    ᯥ譮 ࠦ
஬ ࠣ ᭮ ࠭⢠ ୥ (ਬ, Code Red
-  350000 ࢥ஢).

                 2.3.    " 宫"

     宫 -  ਪ᪨ ४騪   㤮, 
㬥    15         . 
ᥣ  ਭ 祭:  " 饬   騩  
蠭  15  ᫠",   ᯨ࠭ 㭨 ફ . 
뭥   ᢮ ᪠쭮 ⮩ . ॢ ⮩ 
    ਠ 5- ᪠ ୠ "᪨ 蠫".  ஬
⮣,     த,  饭 祭 ࠧ  
Slammer.
        ᬠਢ   ⨬   ⥣    ࠧ
饣   ,   稢襣   ᫮    "
宫".  뫮 ⬥祭,   ࠧ       -
ࠧ   ࠧ    ᮡ  ࠭⢮  IP-ᮢ,  ⮡  
""   ࠧ      .   㯮   ࠩ
  "" 室:
    -           㣮  ᯨ᪠  ࠦ
⮢;
    -     ।    ᫥ ᢮ ⮬ -⢠
( 㥬  ࠦ ࠭⢠ IP-ᮢ).
     室  (  ᭮  ன,  ..    ॠ㥬  
ࠪ⨪)  﫨         ࠧ      ᮮ⢥⢨   
ᯮ樠쭮 .
     .    ᬠਢ 冷 室 IP-ᮢ  
 ᢮ ࠣ ᭮ ࠭⢠.  ᫨  砩,  
   먣   "⮬᪮"  ⥣  ய.      ,
। 㯮 ᫥⥫ ॡ IP-ᮢ?
     ⥬,    ᮢ "⮬᪮" ࠧ ᭮
࠭⢠  砩 冷 ॡ ᮢ.
    ⠪,   ।      .  :)    롮  砩
IP- 㦭 ᯮ짮   :

                      X(i) = (X(i-1)*A + B) mod M.

     ࠢ쭮 ࠭ A    B  横  稪      ᢮
।쭮  祭,    M = 2^32.  -,   ࠡ ⢥
     .    ࠧ...
    祣 譮.        ।  ⢥ ࠢ
᢮ ࢮ ⮬ 砩 ᫮,  ஥    १  M/2
蠣;   ஬   ⮬ - ஥  १ M/4 蠣,   ..
 ⮬,  ࠧ,  ᢮ ॡ ࠢ, ࠢ
祭   稪   १ M/2+M/4 蠣,  _稭__᢮_⥪饣_
_祭_  ..
      뢠  "ࢥ DrMAD-". :)

                   2.4.  c ⨥ ⨢

     ࠪ⨪  ⨢    ࠭    稭  ,
⮬  室  ஬  V(t)    W(t)    U(t)  -   ⢮
楫 / ய祭 ⥬.  㬥, ⥯ N = V+W+U 
1 = v+w+u.
    㤥 ,    楫/ய祭      
  ࠦ.   । ᪮ ࠭ 楫 =
C.   㤥 뢠,    楫        ⮣,  ᪮쪮
஢ 㦥 楫  ⮣,  ந室  砩 ࠧ, 
楫ࠢ,  ய樮쭮 ࠭ ଠ樨   ,
      ࠧ     ᯮ樠쭮.  ⢥⢥,  
楫/ ய祭 設 뢠   "⪨  ".  
    ⠥    ᥡ   ० - 롨 楫 砩
ࠧ. 㬥,  㦥  ⤥쭮 ७樠쭮 ࠢ, 
 ⥬:

    du(t)/dt = C*u(t) -  ᫠ 祭/ய祭
    dv(t)/dt = B*V(t)*(1-B*V(t))-C*u(t) -  ᫠ ࠦ

    㫠 襭      窥 㬥頥  㤮 :),  ⮬
ࠧ 䨪:

                        v ^
                          |       
                          |     
                          |   
                          |  
                          | 
                          +----------------------------->  t

    ..   ᮬ -⠪ ⨣,    - ,
 ⢥     ⭮襭 C/B.  ࠪ⨪ C/B >> 1,  
ᯥ ࠧ ,   祣  ,  ० 祬  稭
᪠뢠 १ 祭.
    ⢥ ,  ᪨ ⮤ ਡ 
᢮ ।.    ॠ쭮   ࠭  
 㣨 , ⨢騥   䠪:
    -    易⥫쭮  ⢥ ;
    - ⨢    ;
    -      饣  ᫠  設          
砥,  砫쭮 ய祭 設;
    - 砥  "䥪⨢"  樭 - ।  設
 ࠧ묨 稭    樭 ;
    -    㬥襭   ᫠   ࠦ   設      
⪫祭 ॣ㦥 ࢥ஢;
    -       த  ࠭⢠  IP-ᮢ,  
ࠧᠭ  ࠢ୮,  "窠";
    -  ..

                        2.5.   

    ० ᥣ        ,      饬  砥  襭
७樠쭮  ࠢ -  ᥬ⢮ ਢ.  뢠  
ࠢ ⥫ ࠭祭 (ࠥ,  ࠭    砫
᫮),      㧨    ᥬ⢮    窠    
⢥ 㭪樨.
     ⮤           ࠢ        
"९⢥ ࠧ".  ࠢ dV(t)/dt=B*V(t) ਢ
( ⮬ ⮣,   dv(0)/dt=B)  ୮ ࠢ (p-B)*Y(p)=1,
㤠    ⭠    㭪    Y(p)=1/(p-B).   믮   ⭮
८ࠧ , 砥 襭 V(t)=1*exp(B*t).
    ࠢ       ⮤  ࠧ ६
/4/.
      ᫨  ᮢᥬ    ࠯  ࠭訪    㬠,  
ࠢ       ᫥,   ਬ,   ⮤    
dv/dt=f(v,t) /3/:

               t(i+1)=t(i)+h
               v(i+1)=v(i)+h*f(v,t)
               i = 0..N
               h = 쪨 蠣 (ਬ, 0.1  0.01).

                        3. 樮 ⮤

    樮 ⮤ ந (筮,   )  ""
ꥪ,   ⥬          ᭮  樨  ⠢
⮢,  ⭮襭,   ࠧ  ६  ..  樮
  ஢:
    1) ନ஢ ᯥਬ (ਬ, ⠪  ஢
⮫ ਪ  來 ⮫);
    2) ᪨  (⭮)  ᯥਬ  (     
뢠 "-૮").
     ⮨⢮ 樮 ⮤ - ,       
 ஢ 祭 ᫮ ⥬  .   ⮪
- ᫮ ᯥ祭 ⮢୮ १.    ,  
樮 ஢  㦭 믮 讥 ᫮ ᯥਬ⮢
("ண"), 祬    ᫮          筮   
⮢୮ १,    祭  /5/.
      㤥    筮,   ⮢୮ ᫮
१.     ᮢ ⢥ ࠪ⨪ - ଠ
ਢ ࠧ .

              3.1. 饥 ᠭ 饣 

     室  ਢ  , ⥬ ,    
ணࠬ,     楫    .            -   
த஢ 騩 室  ஢.
    砫    让 ᨢ稪 楫 ᥫ -  ᭮
࠭⢠.

const
    MAXNOD = 1000;  (* ⢮ 㧫   *)
var
    NODES : array [0..MAXNOD-1] of byte; (* ᭮ ࠭⢮ *)

     㧥  室    ﭨ:

const
    VUL    = 0; (* ஢  梨 *)
    IMM    = 1; (* ஢  梨 (.. 樭஢) *)
    EFF    = 2; (* ࠦ ࢥ *)
    INF    = 3; (* ࠦ ࢥ *)
    OFV    = 4; (* ⪫祭, ⠢ 梨 *)
    OFI    = 5; (* 樭஢,  ᫥ ⪫祭 *)
    OFE    = 6; (* ࠦ ࢥ  ⪫祭 *)
    OFI    = 7; (* ࠦ ࢥ  ⪫祭 *)
     ..

     뫮,  筮,  ॠ  㧥     ꥪ  
᢮ ஬ ਡ⮢  ⮤,   ᨢ稪 㤥  
 100 㧫,   1000000,  ஡ 䥪⨢.

    ⮬ ᤥ      楤ப,  㫨 
⨬ ᨢ:

    InitNodes( nvul,  nimm, neff, ninf : integer) - 樠 㧫
祭ﬨ   ﭨ;
    CheckNodes( var  nvul,  nimm,  neff,  ninf  :  integer)  - ᮡࠥ
ଠ  ﭨ 㧫;
    SearchFor(k1, k2:integer):integer -   ࢮ 襣
㧫   㪠 ⨯;
    Infect(n:integer) -  㪠 㧥;
    Fight(n, me, foe:integer) - 頥  㧫 ࠦ᪨  ᮡ;
    UnlockAndInfect(n:integer) - ᭨ 樭  ࠦ 㧥;
    Cure(n:integer) -  㧥;
    CureAndImmune(n:integer) -  ࠦ 㧥  樭 ;
    Immune(n:integer) - 樭 梨 㧥;
     ..

     ,  ਬ,   룫拉  ⠪ 楤ઠ:

    procedure CureAndImmune(n:integer); (*/樭஢*)
    begin
     if ((n>=0) and (n<MAXNOD)) then
        if ((NODES[n]=INF) or (NODES[n]=EFF)) then NODES[n]:=IMM
    end;

        ਬ   饩   ணࠬ,  ᯮ饩    .
ணࠬ ॠ ᨫ쭮 饭   c  ࠢ묨    묨
᪮ﬨ   ࠧ     ᮢ.  ।,  
騥   ६ (祣  ࠪ⨪ 
  뢠).  ஬  ⮣,        ᪮쪮  㣨 殮.
ਬ,   祭/樭஢  ᠬ    易
   ᫮   ࠭   뫥祭/樭஢   㧫   (  易  
࠭ ଠ樨    ),      ࠧ  ⮦  
ᯮ, ⮬  ਯᠫ    ᠬ "" . ⠪:
 ࠧ, ᪠ ᭮ ࠭⢮ 砭멬 ࠧ, 
 祭/樭஢ ⨢  楫ࠢ, 
100%- 䥪⨢.

 (* () Constantin E. Climentieff aka DrMAD, Samara 2004-05 *)
 ...
 randomize;
 InitNodes(MAXNOD-2, 1, 0, 1);  (* 砫 ﭨ 㧫 *)
 SPEED1:=1; SPEED2:=1;          (*  ࠧ *)
 t:=0;                          (* 쭮 ६ *)
 repeat
  CheckNodes(n0,n1,n2,n3);
  for i:=0 to maximum(n1-1,n3-1) do begin
   if i<=n3 then for i1:=1 to SPEED1 do Infect(random(MAXNOD));
   if i<=n1 then for i2:=1 to SPEED2 do CureAndImmune(SearchFor(INF, VUL));
  end;
  t:=t+1;
 until not Finish;
 ...

                       3.2.  १

     ਢ   १  ஢.    ॠ
ᯥਬ  뢠 ࠧ 襥 ⢮ 䠪஢ (.  .
2.4),    䠪஢ ᢠ ॠ쭮 ७   
ᯥਬ  祭,    ⮬   㣨 䠪ࠬ 
ਢ 樮 १  .   (ࠪ᪨
⮤  )  業 祭  䠪஢:  㤠 
⮢,   ࢮ砫쭮 ⢮  ࠦ  設,  
㧫     ⪫/ய      ।,  ⮡  
  ஢  .
    ਢ  १  ࠪ᪮  ᫠      
,    㦠 ⮫쪮  樨.  ,  ᢥ 
ࠢ  㣫 90 ࠤᮢ. :)

     1: ⪠ , ᯮ樠쭮 ࠧ .

      1.0% 
      3.0% 
      7.0% 
     15.0% 
     31.0% 
     63.0% 
    100.0% 

     2:  ࠧ  "᪮" 

      1.0% 
      3.0% 
      7.0% 
     14.0% 
     27.0% 
     49.0% 
     69.0% 
     85.0% 
     95.0% 
     99.0% 
    100.0% 

     3: ⨢ 䥪⨢ /樭 設, ࠦ
"᪨" ࢥ.

     1.0% 
     1.0% 
     1.0% 
     2.0% 
     4.0% 
     7.0% 
    13.0% 
    23.0% 
    37.0% 
    58.0% 
    72.0% 
    80.0% 
    76.0% 
    56.0% 
    13.0% 
     0.0% 

     4:  ᮢ.  ࠧ   ᪮
       ⮬    㡨 ࠣ,    ᮡ.
  ᮮ⢥,  ਬ,  졥  ᮢ  Yaha    Sahay
(   -   ଥ  Gigabyte).  ᪮  襭    
᪮   ᮢ  ⬥⪨ 50%,      襭  -
⮩稢.    ࠪ⨪  ᨬ쭮      筠
   ᮢ,  ⥫쭮  ஢ 50%.  
⢥ঠ 樮 ஢:

                                 ன

 1.0%                          1.0%               1.0% 
 2.0%                          2.0%               2.0% 
 4.0%                         4.0%              4.0% 
 7.0%                         8.0%             8.0% 
12.0%                      14.0%           15.0% 
18.0%                     21.0%         24.0% 
27.0%                   33.0%      35.0% 
42.0%               38.0%     47.0% 
52.0%            48.0%  48.0% 
60.0%          45.0%   43.0% 
67.0%         43.0%    47.0% 
71.0%        46.0%   45.0% 
76.0%      48.0%  44.0% 
79.0%      50.0%  40.0% 
81.0%     47.0%   34.0% 
85.0%    47.0%   22.0% 
89.0%   49.0%  25.0% 
92.0%  50.0%  16.0% 
95.0%  47.0%   12.0% 
93.0%  45.0%    7.0% 
94.0%  43.0%     4.0% 
93.0%  41.0%     3.0% 
94.0%  49.0%   3.0% 

     5:           ࠧ  ᪨,  
  ⮫쪮 ,   樭 設.

     1.0% 
     2.0% 
     4.0% 
     8.0% 
    15.0% 
    26.0% 
    42.0% 
    59.0% 
    67.0% 
    75.0% 
    70.0% 
    55.0% 
    36.0% 
    26.0% 
    18.0% 
    17.0% 
    15.0% 
    12.0% 
    10.0% 
    10.0% 
    10.0% 
    10.0% 
     9.0% 
     8.0% 
     5.0% 
     5.0% 

             3.3. 楯 : 訫  

    ਢ १  ᭮뢠 ᫥騩 ⥧:   -
 ⠪    ୠ⨢ ࠤ樮  ⮤ ⨢⢨
ࠧ.   ࠩ  ,     ମ  ஢
ࠧ⨥ ࠧ,    㤥 祭    ࠤ樮묨
।⢠.
     ⨥ "" 易  ⥫⢠.
    -,           ᨬ쭮    ।,
㦨       㬥   "ᠬ뢠"     祭
- ।  ப         室   -
ப⥫쭮 ᨣ.
    -,      ᨬ쭮 ࠧ  ਠ,
,  ⮡      ଥ  ,  ᯮ騩   
    ⥪騩  ન  ,   ⮬ ᫥  㭨,
  ⮫쪮  ࠧࠡ稪.    易    ⥬,      ࠭
⭮,  ⨢         ணࠬ-⭮ ଥ
ਤ  .  ஬  ⮣,        ।⢠    
ᨬ쭮  ਬ⨢,      㭨ᠫ:  ਭ㤨⥫쭮
⨥ ⮢, 㡨⢮ ࠦ᪨ ᮢ  , 饭  
॥  ..   ન    ᬮ,     ન 
ࠧࠡ  ࠧࠡ稪  ॡ  ६,      
᪠   ᢮        ਧ .
    , ࠧࠡ⪮ ࢥ      ᪫⥫쭮
᮪ࠫ    ᮪஢  ,      
  ᫮ ஦襩  ᯨ樨.  㬥,  ࠢ⥫⢠,
ᯥ㦡,  ⨢  ,  ।⢠  ᮢ  ଠ樨  
    ࠧ묨 ।     ࠢ  
ࠧࠡ稪 ᥬ 묨 ᯮᮡ.  ⮬   ⮢.
    ,   ॠ  ᫮    ।⠢  ਥ묨  
" ",  ᮢᥬ 㣨   ᯮ짮 ࢥ.
    -訫: ᪠   ᪮ (ਬ,  १  
)  ᫥ 㡫 -  ન  . 讣
ࡠ    稭,    㣠,    㦤  ⮢   
३ ࠧࠡ ,  ஢ - ३  ᯮ짮.
    -: ᪠      १      ࢠ
६ (ਬ, १ 2-3 ) ᫥ 㡫   થ.
ய祭 設  㭭    ⮬  ,    ய祭
 樭஢ ᮬ ਭ㤨⥫쭮.
      ࠭    ।  ஫㥬
猪, 祬 ⮬       १ 誨
"" ,  ᠭ ᮢ襭⭨ ਤમ  ࠡ᪨
⮬.

                              

    1.  ..     ᯮ浪.  - .: 㪠, 1982. -
176 .
    2. 祭    .    ࢮ:    稭        ᫥⢨.
http://az13.mail333.com/4/vh6.htm
    3. 类 ..  ⥬ MATHCAD.  - .:   , 1993 - 128
.
    4. ன⥩  ..,  異  ..  ࠢ筨  ⥬⨪ 
஢   . - .: 㪠, 1986. - 544 .
    5. ⮢ ..,   ..  ஢ ⥬.  - .:  .
., 1998. - 319 .
